Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of C9⋊A4 |
| automorphism | Aut(G) = ((C3⋊D3)⋊C3)⋊S4 (order 1296) · Inn(G) = C3×A4 · Out(G) = D32 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | C22⋊H(3,𝔽3) |
| unambiguous semidirect product | C32⋊A4 |
| unambiguous semidirect product | (C3×A4)⋊C3 |
| Fitting | C22×C32 ⋊ C3 |
| Sylow | C22 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×A4 |
| commutator, socle, cosocle | C22×C3 | C32 |
| Fitting | C22×C32 | C3 |
| hypercenter | C32 | A4 |
| hypocenter | C22 | H(3,𝔽3) |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C3 ◁ C32 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,4)(2,3)(6,9,10)(8,12,13);(2,3,4)(5,6,8)(7,10,12)(9,13,11)Polycyclic presentation:
3^3,2^2|b^a=bc,d^a=e,e^a=de20 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 22);